The key thing to remember is that chivalric culture in the Late Middle Ages went in two quite paradoxical directions. On the one hand, chivalric culture was becoming more elitist as formal knighthood was becoming restricted to a smaller pool of men than it had been in the twelfth and thirteenth centuries - c.1200 was probably when the peak number of knights in Western Europe was reached (an estimated 4,000 in Angevin England with its population of around 3.1 million people and no doubt substantially more in the more populous Kingdom of France, the German Empire and the Italian city communes). As the thirteenth century had worn on, a combination of various economic pressures such as galloping inflation driving down the value of rents on manorial estates, the growing scarcity of new land save for in Christian Spain and on the eastern frontiers of Latin Christendom and the growing costs of adequately equipping a knight for war. Between 1150 and 1225, knights went from wearing just knee-length mail hauberks and simple iron open-face helmets to being clad from head to toe in mail, and after c.1180 the shift from open-face helmets (like the classic conical nasal one believed to be ubiquitous in the tenth to twelfth centuries thanks to the Bayeux tapestry, the Wenceslas helmet in Prague and the Moravian helmet in Vienna) to masked and, no later than c.1220, fully enclosed helmets - the great helm or the classic "crusader helmet" as most people know it from popular culture. At the same time, heraldry came in (itself a marker of growing status and exclusivity among knights) and that meant that a lot had to spent in procuring surcoats and shields bearing one's new personal/ family coat of arms. Then, from c.1240, as clearly indicated by the statue of St Maurice (a Roman legionary from Egypt martyred in the 3rd century AD) at Magdeburg Cathedral produced around this time, knights started to wear coats of steel plates sewn together underneath their surcoats and over their mail hauberks. By the early fourteenth century, we see more plate armour coming through i.e. the monumental brass of Sir William FitzRalph (died 1323) from Pebmarsh parish church in Essex, shows him wearing plate greaves on his legs, on his forearms plate vambraces and on his upper arms plate spaulders. Quite similar features can be noted in the painting of the mercenary captain Guidoriccio da Fogliano painted by the early renaissance artist Ambroggio Lorenzetti and commissioned by the republican government of Siena in 1327. When we get to the 1360s, we can see full plate armour emerging, as is visible from something like the tomb sculpture of the great English knight Sir Reginald Cobham (died 1361) in Lingfield parish church, Surrey. Then, after c.1400 plate armour becomes more articulated and elaborate than ever before, to the point that if you're going to get the most protective and stylish kit you need to go over to one of the principal armour factories in places like Milan (Italy) or Augsburg (Germany) to have armour made specifically to fit your figure like a carefully designed exoskeleton/ second skin and have whatever decorative features you might desire (large spaulders, gilded bits, fluting etc). So basically over the course of the Later Middle Ages the cost of kitting out a knight for war drastically increases, with accelerating pace after the first quarter of the fourteenth century. This made the military life increasingly difficult to afford for a lot of lesser noble landowners like much of the English gentry, who by the fourteentb century were increasingly turning to civilian professions like law and accountancy. But it wasn't just about the costs of knighthood. The state-building activities of the Plantagenets in England and the Capetians in France meant that knighthood came with significant administrative burdens as well - in England knights were by the thirteenth century obliged to sit on juries, serve on commissions of the Peace, perform various local magistracies and even stand as elected representatives in Parliament, while in France they acted as royal bailiffs, seneschals and inquirers into corruption and maladministration in local government. Thus it became the strategy of many lesser landowners to wilfully avoid becoming knights to avoid the responsibilities they could be liable for if they accepted dubbing i.e. in England, any male landowner owning estates worth more than £40 was obliged to be dubbed a knight, but many refused and the royal chancery frequently issued summons and fines to the members of the gentry who tried to evade knighthood. The ceremony of being dubbed a knight, first mentioned in the Old French poem Ordene de Chevalerie (written c.1225), was also incredibly time-consuming and expensive - a knight had to stay in a perfumed bath for a long time, then sleep on a bed of clean linen, then dress in a red tunic (symbolising readiness to shed blood for the faith), black stockings (symbolising human mortality), a white belt (symbolising sexual purity and chastity) and a red cloak to symbolise humility, then he had to stand for a vigil in a church for a whole night then the next morning attend mass, be given the gilded spurs and the belt and then tap with a sword on both shoulders and a kiss by the man conferring the knighthood. The various secular (as opposed to religious ones like the Templars and Hospitallers) orders of chivalry founded by kings and princes in the fourteenth and fifteenth centuries, devised for themselves even more expensive and elaborate ceremonies. As a result, many noblemen who wanted to become knights would avoid the regular dubbing ceremony and instead would try and get dubbed on their first military campaign i.e. at the beginning of the Crecy Campaign in 1346, Edward III dubbed dozens of young men as knights when they landed in Normandy as an act of largesse (generosity); and many knights would get dubbed after battles for personal bravery i.e. Sir John Hawkwood being knighted by the Black Prince immediately after the battle of Poitiers. Combine that also with the fact that sovereigns (kings, princes, republican communes) were increasingly beginning to assert monopolies on the dubbing ceremony and you get a situation in which many nobles would get dubbed relatively late in life (well past the standard age of 21) if at all i.e. the great French knight Jean le Maingre (1366 - 1421), nicknamed Boucicaut, getting dubbed aged only 16 after defeating the Flemish urban militias at Roosebeke in 1382 (note him being dubbed after a battle for personal bravery) was unusual. The celebrated French knight Geoffroi de Charny (1304 - 1356), a well-to-do but not fabulously wealthy noble warrior who wrote three works on his profession including the so-called Livre de Chevalerie (Book of Chivalry), was not knighted until 1343, by which point he was 39 years old, was perhaps more typical than Boucicaut.
So, what you had by roughly the time when the last duel is set was a very small club. For example, in the first phase of the Hundred Years' War (1337 - 1360) only around 12% of the French cavalry were actually knights (about 3,700 in all) while more than 87% of them were squires or plain men-at-arms. While in England, where the gentry on the whole tended to be richer than the lesser nobility of France (thanks to a combination of prudent inheritance strategies and more efficient and attuned to market forces estate management), the proportion of knights was greater i.e. 25% of the cavalry on the Black Prince's Poitiers campaign of 1356. But by the time of the Battle of Agincourt in 1415, whose anniversary is today, only 8% of the English cavalry brought on that campaign were knights. Thus, knighthood in the mid-fourteenth century really was a badge of exclusivity, far from being synonymous with nobility like it had been a century earlier. Some historians have noticed a tendency among knights to try and zealously safeguard knighthood for a small elite core of devoted warriors, albeit of varying social backgrounds, whilst trying to exclude those who pursued the demands and ethos of knighthood with less rigour or were essentially civilian in outlook. For example, the French knight Geoffroi de Charny frequently writes in his Livre de Chevalerie about his disdain for knights who live effete, luxurious, pampered lifestyles and aren't constantly striving to attain greater honour and renown through displays of prowess, courage, generosity and courtesy. From a somewhat different angle, the fourteenth century Florentine poet, Francesco Sacchetti complained about merchants, lawyers and bankers being given knighthoods (this had in fact been going on since the twelfth century in Italy, given that these sort of middle class citizens had then been obliged to serve as cavalry in the militia, which by the fourteenth century had largely been farmed out to mercenaries) by saying that they might as well confer them on oxen and donkeys than these comfortable civilian types.
Yet as I noted earlier, there was another direction that chivalric culture went in. That was the tendency for chivalric culture to become more inclusive than ever. For example, Geoffroi de Charny addresses his target audience collectively as men-at-arms, not as knights, and while he holds up formal knights as most aspirational his concept of the prudhomme (worthy man) does not factor in titles or status). Indeed, as already referenced in the first part of my answer, a very large percentage (indeed the majority) of the heavily armoured cavalrymen who served in late medieval armies were not knights but squires and men at arms, yet they of course belonged to similar social backgrounds and had the same culture and values as those who were formal knights, and could hope to get knighted on campaign if their commanders were feeing generous or if they performed exceptionally well.
I've also alluded to lawyers, bankers and merchants, men who had made their careers through civilian service, getting knighted and that was common in England, France and communal Italy as knighthood was an extremely prestigious award to give anyone for loyal service to the state, even if none of it entailed fighting, and it would help them ease themselves into the ranks of the social and political elite. For example, William de la Pole (1290 - 1366), a wool merchant from Hull who acted as chief banker and Baron of the Exchequer to Edward III was knighted for his service to the English state, and his descendants would go on to be earls of Suffolk and, by the 1440s, the king's chief ministers. From the 1280s, the French kings similarly issued patents of nobility and knighthoods to the lawyers and professional administrators who served them well.
What you also get is a reverence of chivalric culture penetrating all levels of society. By the fifteenth century you have companies of archers forming co-fraternities in imitation of orders of chivalry, yeomen farmers and other rich peasants acquiring coats of arms and chivalric romances like Thomas Malory's Le Morte d'Arthur (1485) being read by merchants, lawyers, bankers, artisans and yeomen. Similarly, many genteel English lawyers and members of parliament as late as the early seventeenth century would depict themselves in their tomb monuments as decked out in suits of armour even if they'd never served on a military campaign or participated in a tournament in their entire lives
What you ultimately got out of this was a society that revered knighthood and its culture and values even as the number of formally dubbed knights dwindled and some of those among them tried to close ranks. Late medieval knighthood was a highly respected and prestigious, if very exclusive club, though the routes into it varied a lot and what it meant to its members themselves did similarly.
For further reading, I would insist you read the chapter "Chivalry beyond formal knighthood" in Richard Kaeuper, "Medieval Chivalry", Cambridge (2015). Kaeuper focuses on precisely what you are interested in and gives very comprehensive, balanced coverage and in the footnotes provides plenty of citations through which you can explore this subject even further.
Arguably the single most important factor of knighthood was the adherence to the chivalric code, and we can see how heavily the concept of chivalry was emphasised in late medieval society through contemporary authors such a Chretien de Troyes. By the C14th, chivalry had come to be primarily associated with a particular fighting style, and with the innovation of this fighting style came the need for specialised training in it, and as this training required equipment such as a warhorse and a suit of armour, it become almost entirely exclusive to the elite military branch of the aristocracy, knights.
In order for a knight to be considered adequately chivalric, he needed to fulfil a variety of criteria when it came to his character. These criteria included reaching the expected level of proficiency in values such as piety, chastity, humility, love, and courtly behaviour, but most importantly in prowess. Prowess is the display of one’s martial abilities and courage on the battlefield, and as Richard Kaeuper states, it was so highly regarded by knights due to the fact that committing feats of prowess was one of the most effective methods of gaining honour. Jean Froissart once stated that “As firewood cannot burn without flame, neither can a gentleman achieve perfect honour nor worldly renown without prowess.”
In terms of the significance, prestige and politics of being knighted, it essentially advertised you as the ideal masculine male in the order of those who fight. It meant that you would have had developed the highest standard of courtly behaviour which would have allowed them to find and be able to court a female lover. However it would also have been an immediate indicator of your wealth, as you had been able to afford to train in the chivalric fighting style, and reaching the point where you are knighted would let everyone know you had immense levels of prowess on the battlefield.
If you were to never advance beyond being a squire, it could be because you are lacking in at least one area of the aforementioned values (piety, chastity, humility, love, courtly behaviour and/or prowess), or it could simply be a case of being too young or not having had the opportunity to display your battlefield prowess or to commit some heroic deed.
We can see some of the differences in how knights were viewed by late medieval society as opposed to squires in the opening to their chapters in Chaucer's The Canterbury Tales.
“There was a knight, a most distinguished man,
Who from the day on which he first began,
To ride abroad had followed chivalry,
Truth, honour, generousness and courtesy.
He had done nobly in his sovereign’s war
And ridden into battle, no man more,
As well in Christian as in heathen places,
And ever honoured for his noble graces.”
For the knight, and
“He had his son with him, a fine young squire,
A lover and cadet, a lad on fire…
And had done valiantly in little space,
Of time, in hope to win his lady’s grace…
He loved so hotly that till dawn grew pale
He slept as little as a nightingale.”
For the squire.